Gear arrangement for providing an oscillating rotational motion

ABSTRACT

A pair of meshed gears each of which comprises a plurality of sectors having the configuration of a logarithmic spiral provide an oscillating motion with respect to each other as they rotate. The gears can be coupled to the rotors of rotary piston machines and the like to provide the desired oscillating rotational motion thereto.

United States Patent 11 1 McMahon [451 May 1,1973

[ GEAR ARRANGEMENT FOR PROVIDING AN OSCILLATING ROTATIONAL MOTION [76]Inventor: William McMahon, 20 Hillside Ave., Summit, NJ. 07901 [22]Filed: Feb. 14, 1972 21 Appl. NO; 226,127

4 74/437 [51] Int. Cl. ..F0lcl/00,F16h35/()2,F()4c1/00 [58] FieldofSearch ..418/3336, 150; 74/393,437; 123/8,47

[56] References Cited UNITED STATES PATENTS 8/1959 Kitano ..74/4373,061,180 10/1962 Durgin ..4l8/36 3,112,062 11/1963 Way 3,398,643 8/1968Schudt 3,430,573 3/1969 Groeger ..41 8/36 Primary ExaminerCarlton R.Croyle Assistant Examiner-John J. Vrablik Attnrney-Alvin D. Hooper I 57]ABSTRACT A pair of meshed gears each of which comprises a plu rztlity ofsectors having the configuration of a logarithmic spiral provide anoscillating motion with respect to each other as they rotate. The gearscan be coupled to the rotors of rotary piston machines and the like toprovide the desired oscillating rotational motion thereto.

16 Claims, 9 Drawing Figures Patented May 1, 1973 3 Sheets-Sheet 1Patented May'l, 1973 3 Sheets-Sheet 2 Patented May 1, 1973 3,730,654

3 Sheets-Sheet 3 GEAR ARRANGEMENT FOR PROVIDING AN OSCILLATINGROTATIONAL MOTION BACKGROUND OF THE INVENTION adjacent pistons alwaysrotate in the same direction but with alternating increasing anddecreasing speeds with respect to each other.

Various gear arrangements have been disclosed in the prior art forproviding an oscillating rotational motion. These arrangements haveprimarily utilized sector gears and elliptical gears. However, thesetypes of gears are not ideally suited for such applications. Theintermittent meshing of sector gears resultsiin high stresses and noise.These gears must be manufactured with very tight tolerances to preventexcessive wear which would increase the noise and stresses duringmeshing. When elliptical gears are used, the point of meshing does notremain on the line joining the axes or centers of two meshing ellipticalgears. Thus the gears tend to unmesh or uncouple. Further, meshing awayfrom the line joining the axes can produce high stresses in the gears.

Another disadvantage of the gear arrangements previously disclosed isthe number of gears required and the resulting complexity of thearrangements.

SUMMARY OF THE INVENTION The foregoing disadvantages are overcome inaccordance with this invention by a gear arrangement which utilizesmeshed gears having sectors with configurations ofa logarithmic spiralfor obtaining an oscillating rotational motion. Each gear has fouridentical quadrants each of which has a varying pitch diameter definedby a logarithmic spiral. The gears are meshed so that the mesh point isalways on a line joining their axes and the rate of rotation of thegears alternately increases and decreases with respect to each otherwithin every 90 of rotation, i.e., within every quadrant of the gears.The gears can be connected to the rotors of rotary piston machines andthe like to provide an oscillating rotational motion for definingalternately expanding and contracting chambers therein.

BRIEF DESCRIPTION OF THE DRAWING The invention will be more fullycomprehended from the following detailed description and accompanyingdrawing in which:

FIG. 1 is an exploded perspective view partly in section of a rotarypiston machine utilizing a gear arrangement in accordance with thisinvention;

FIG. 2 is an elevation view of the logarithmic spiral gear utilized inFIG. 1;

FIG. 3A, 3B and 3C are schematic representations of the operation of thegears of FIG. 1;

FIG. 4A, 4B and 4C are schematic representations of the configurationsof the machine of FIG. 1 with reference to the positions of the gears inFIG. 3A, 3B and 3C, respectively; and

FIG. 5 is a perspective view of a second gear arrangement utilizingintermediate gears to obtain different rotational speeds between thelogarithmic gears and the associated rotors.

DETAILED DESCRIPTION The gear arrangements of this invention will bedescribed with particular reference to rotary piston machines. However,it is to be clearly understood that the gear arrangements can beutilized anywhere an oscillating rotational motion is desirable.

In FIG. 1 is shown partly in section an exploded perspective view of arotary piston machine 101 comprising two coaxial rotors or shafts 2 and4 about each of which a plurality of vanes 6 and. 8, respectively, aremounted at substantially equal angular intervals. A plate or flange ismounted on one end of shaft2 to which vanes 6 are mounted so that theedges 12 of vanes 6 are spaced asmall distance 14 from shaft 2. Vanes 8are mounted directly upon shaft or rotor 4. Flange 10 can have a seriesof ports or openings 11 therein which function as will hereinafterbecome apparent. Vanes 6 and 8 and rotors 2 and 4 are enclosed within ahousing 16 which in the illustrative embodiment comprises a generallycylindrical container with one open end. Housing 16 is mounted to theouter peripheries of vanes 8 and rotates therewith.

Shaft 2 fits within shaft 4 and can be journaled therein by well knowntechniques. Plate 10 fits against and seals the open end of housing 16.Vanes 6 fit between vanes 8 in an alternate configuration around thecircumference of shaft 4. The edges 5, 7, 9 and 12 of vanes 6 and 8 formseals where appropriate with corresponding portions of shaft 4, plate 10and housing 16 so that eight independent chambers are defined.

Shaft 2 extends from the opposite end of shaft 4 and has fastenedthereto a logarithmic spiral gear 18. Gear 18 can be coupled to a powersource or power utilizing apparatus depending upon whether machine 101is being used as a pump or an engine.

Coupled or meshed with gear 18 is another logarithmic spiral gear 20.Gear 20 is connected through shaft 23 and identical intermediate gears22 and 24 to shaft 4 so that shaft 4 rotates at the same rate but in theopposite direction as gear 20, i.e., shafts 2 and 4 rotate in the samedirection. Gears 22 and 24 can comprise circular spur gears well knownin the prior art.

FIG. 2 is an elevation view of a logarithmic spiral gear like gears 18and 20 utilized in machine 101. Gear 30 has four identical sectors orquadrants 32, 34, 36 and 38 which are joined as shown to form a roughlybow-like" configuration. By designating gear 30 as a logarithmic spiral,it is meant that any point 42 along the pitch diameter of eachquadrantis defined by the function: D=e" where D equals the radius 44 from thecenter 46 of gear 30 to point 42; e is the base of the naturallogarithm; a is the angle 48 between the radius 50 to the starting point52 of pitch diameter 40 and the radius 44 to point 42; and k is aconstant determined by the dimensions of gear 30 such as starting point52 and end point 54 of pitch diameter 40. The distance 56 along thepitch diameter 40 from point 52 to point 42 is given by: S=(D1) (l+l/kwhere S equals distance 56.

The teeth 60 and spaces 62 of gear 30 are formed about radii 44 by knowntechniques. With respect to radii 44 teeth 60 have the approximate shapeof spur gear teeth with one side cut slightly deeper than the otherside. The width 64 of teeth 60 and width 66 of spaces 62 aresubstantially equal along the pitch diameter 40.

In order to combine four logarithmic spiral quadrants into a gear 30 asshown, it is necessary that the combined number of teeth 60 and spaces62 in each quadrant be an odd number. This is required in order thateach quadrant have a half-tooth at point 52 and a half-space at point54, or vice versa, which combines with the corresponding half-tooth orhalf-space from the adjacent quadrant to form a complete tooth 60 orspace 62. Also the number of teeth 60 in any quadrant must equal thenumber of spaces 62 in that quadrant. The actual number of teeth 60 andspaces 62 utilized in each quadrant; and accordingly the tooth size, isdetermined by known gear design techniques which depend upon suchfactors as the forces being transmitted by the gears, etc.

The operation of two meshing logarithmic spiral gears such as gear 30,and accordingly the operation of the vanes or pistons on the rotorsassociated therewith, is illustrated in FIG. 3A, 3B and 3C and FIG. 4A,4B and 4C, respectively. In FIG. 3A two logarithmic spiral gears 80 and82 are oriented at 90 degrees with respect to each other and coupled ormeshed. The position of vanes 90 A, B, C, D and 92 A, B, C, D associatedwith gears 80 and 82, respectively, are shown in FIG. 4A. For example,gear 80 and vanes 90 A, B, C, D can correspond to gear 18 and vanes 6,respectively; gear 82 and vanes 92 A, B, C, D can correspond to gear 20and vanes 8, respectively, and axis x--x corresponds to axis yy ofFIG. 1. At this position which can be considered an initial or startingposition the vanes 90 A, B, C, D and 92 A, B, C, D are equally angularlyspaced around the circumference of shafts 94 and 95 thereby definingeight equal chambers 96 A, B, C, I-I. When gears 80 and 82 rotate asshown, gear 82 initially rotates at a faster rate than gear 80 untilsome intermediate point illustrated in FIG. 3B is reached. At this pointthe angle of rotation 84 of gear 82 is greater than the correspondingangle of rotation 86 of gear 80. Thus, as shown in FIG. 4B vanes orpistons 90 A, B, C, D and 92 A, B, C, D associated with gears 80 and 82,respectively, have also rotated different amounts, i.e., 92 A, B, C, Dhave rotated more, thereby causing some chambers 96A, C, E, G to expandand others 96 B, D, F, H to contract or compress.

As gears 80 and 82 rotate beyond the intermediate point shown in FIG.3B, gear 80 now rotates faster than gear 82 until the positionillustrated in FIG. 3C is reached at which point both gears have rotatedan equal amount of 90". As shown in FIG. 4C, during this segment ofrotation pistons 90 A, B, C, D and 92 A, B, C, D have correspondinglyreversed their relative speeds of rotation so that chambers 96 A, C, E,G have contracted and chambers 96 B, D, F, II have expanded so that allchambers have returned to their original sizes, i.e., the sizes shown inFIG. 4A but rotated 90 degrees therefrom. Upon continued rotation pastthe position shown in FIG. 3C, gear will continue to rotate faster thangear 82 until another intermediate point similar to FIG. 3B is reachedat which point the relative rates of rotation again reverse. Accordinglyduring this continued rotation chambers 96 A, C, E, G continue tocontract and chambers 96 B, D, F, H continue to expand until theintermediate point is reached at which the expansion and contractionagain reverses.

From the foregoing, it is apparent that each chamber 96A, H contractsand then expands, or vice versa, the same amount with respect to itsinitial or nominal size indicated in FIG. 4A That is, with the indicatedgear arrangement a chamber will contract and return to its beginning orinitial configuration during one segment of rotation. During thesubsequent 90 segment of rotation the chamber will expand and return toits initial configuration. Thus each chamber undergoes a completeexpansion-contraction cycle during each of rotation.

The relative amount of contraction and expansion of chambers 96A, H withrespect to their initial sizes depends upon such parameters as thedimensions of gears 80 and 82, the angular thickness 97 of vanes orpistons 90A, D, and 92A, D, and the gear arrangement connecting gears 80and 82 with the vanes. When chambers 96A H define combustion chamber foran engine or pump chambers for a pump, it is desirable that the volumeof a chamber be reduced essentially to zero during the contraction partof the cycle. Thus as shown in FIG. 4B chamber 968 has been reducedessentially to zero and the vanes 92A and 90B defining this chamber areessentially touching at the end of the contraction phaseof the cycle.The angular thickness 97 of the vanes is determined by such factors asthe strength required to transmit the forces generated or expended inthe chambers. For purposes of illustration, vanes 90A, D, and 92A, Dmight have an angular thickness 97 of approximately 15. Thus in theconfiguration shown in FIG. 4A, the nominal or initial size or thickness98 of a chamber would be 30, i.e., eight vanes having a thickness of 15each and eight chambers having a nominal size of 30 each equals 360.Accordingly, in order to reduce a chamber to zero volume during itscontraction part of the cycle, the gear arrangement must provide arelative oscillating motion of 30. This also means that a chamber willhave a maximum angular thickness 98 of 60 during the expansion part ofthe cycle, i.e., 30 expansion above its nominal size of 30.

In order to provide approximately 30 relative motion between gears 80and 82 during each quadrant or 90 of rotation, it can be establishedmathematically that the major axis of the gear quadrant in logarithmicspiral gears 80 and 82 must be approximately four times the minor axisthereof. Referring to FIG. 2, this means that radius 58 is four timesgreater than radius 50. These dimensions also establish the value of theconstant k, previously discussed, as approximately 0.8825 as determinedfrom the boundary conditions: D=l when the angle a=0 and D=4 when theangle a=90With the value of k thus established, the coordinates of anypoint along the pitch diameter 40 can be readily determined by theequations previously given.

As previously discussed, with the gear arrangement shown in FIG. 1, eachchamber undergoes a complete expansion-compression cycle during each twoquadrants or 180 of rotation. Thus if machine 101 were used as atwo-cycle engine, two two-stroke cycles could be obtained from eachcylinder or chamber during each 360 of rotation. The expansion andcontraction of the chambers could be synchronized, as is known in theart, with the passage of openings or port therein by appropriate fuelinlet ports, intake and exhaust ports to provide a fuel mixture to thechambers, ignite such fuel mixture, and exhaust the gases, respectively.For example, one of the openings 11 communicates with each chamberthroughout the entire expansion-contraction cycle and could be used assuch opening or port for the chamber.

The gear arrangement shown in FIG. 1 would provide only one fourstrokecycle for each 360 of rotation. However, it is often desirable to have asymmetrical or balanced machine which provides a complete four-strokecycle every 180. This can be accomplished as shown in FIG. 5 by addingintermediate gearing between the logarithmic spiral gears 18 and and theshafts 2 and 4, respectively. Shaft 2 has a circular gear 106 on the endthereof which is connected through intermediate circular gears 104, 103and 102 and shaft 107 to logarithmic spiral gear 18 in such a way as togive a two-to-one reduction between gear 18 and shaft 2, i.e., shaft 2rotates at half the rate of gear 18. Likewise, intermediate circulargears 108 and 110 give a two-to-one reduction between gear 20 and shaft4. Thus for each complete revolution of gears 18 and 20, the chamberswould undergo two complete expansioncontraction cycles, i.e., onefour-stroke cycle, while rotating only 180. Chambers utilizing the geararrangement of FIG. 5 will expand or contract only onehalf as much asthose in FIG. 1. Accordingly, additional chambers could be added, thethickness of the vanes could be increased, or the design of the gears 18and 20 could be changed as previously discussed to insure that thevolume of a chamber is essentially reduced to zero during thecontraction or compression part of the cycle.

From the foregoing, it is clear that the number of vanes and the gearingbetween the logarithmic gears 18 and 20 and rotors 2 and 4 can be variedto obtain varia tions in the operation of machine 101. However thelogarithmic spiral gears 18 and 20 provide the basic oscillating motionrequired in all such variations. The design of these logarithmic spiralgears 18 and 20 can be varied as previously described to provide a widerange of magnitudes in such oscillating motion. The logarithmic spiralgears remain smoothly meshing at all times thereby avoiding the highstresses and noise caused by the intermediate meshing of sector gears.As shown in FIG. 3A, B, C, the mesh point 87 of the logarithmic gearsalways remains along the line 88 joining the centers of the gearsthereby decreasing the likelihood that such gears will become unmeshed.

Although the operation of the logarithmic spiral gear arrangements ofthis invention have been described with particular reference to rotarypiston machines, it is to be clearly understood that the applications ofthe invention are not to be limited to such machines. The geararrangement is intended to have application wherever an oscillatingrotational motion is needed.

What is claimed is:

1. A pair of meshed non-circular gears each of which has a plurality ofgear sectors, each of said sectors having a pitch diameter defined by alogarithmic spiral, said gears being meshed so that said gears contacteach other at different relative points along said respective pitchdiameters as said gears rotate whereby one of said gears alternatelyrotates at a faster and then a slower rate than the other of said gears.

2. A pair of meshed non-circular gears for causing a first rotorconnected to one of said gears alternately to rotate at a faster andthen a slower rate than a second rotor connected to the other of saidgears thereby to produce an oscillating rotary motion between saidrotors, each of said gears having a plurality of identical sectors eachof which has a pitch diameter defined by a logarithmic spiral of theform:

where: D is the radius from the center of said gear to any point alongsaid pitch diameter;

e is the base of the natural logarithm;

a is the angle between the radius to the starting point of said pitchdiameter and said radius D; and

k is a constant determined by the ratio of the radii to the ending pointand said starting point of said pitch diameter, said gears being rotatedwith respect to each other before meshing so thatsaid gears contact eachother at different relative points along said pitch diameters as saidgears rotate, whereby said oscillating rotary motion is obtained.

3. Apparatus in accordance with claim 2 wherein said ratio of said radiiis four whereby said oscillating motion of said rotors with respect toeach other is approximately 30.

4. Apparatus in accordance with claim 2 wherein the number of teeth andtooth spaces in each of said sectors is the same and each is an integerplus one half so that adjacent ones of said sectors can be joined toform a continuous set of teeth and spaces.

5. Apparatus in accordance with claim 2 including intermediate gearsinterconnecting said non-circular gears with respective ones of saidrotors so that said ro tors can rotate at speeds different than saidnon-circular gears.

6. Apparatus in accordance with claim 2 wherein said gear has foursectors forming four gear quadrants, said quadrants being symmetricalabout each of two orthogonal axes through the center of said gear, saidgears being meshed so that one complete oscillating cycle of said rotorsis obtained as said gears rotate through two of said quadrantscorresponding to of rotation.

7. Apparatus in accordance with claim 6 wherein 'said gears are rotatedwith respect to each other so that said starting point of one of saidgears meshes with said ending point of the other of said gears.

8. In a rotary piston machine including a housing, a pair of coaxialrotors within said housing, and an equal number of pistons connected toeach of said rotors for movement therewith within said housing, saidpistons on said rotors alternating about the circumference of saidhousing and defining a plurality of chambers therein; means forinterconnecting said rotors and adapted to cause adjacent ones of saidpistons to alternately approach and recede from each other as saidrotors rotate within said housing thereby to cause said chambers toalternately contract and expand, including:

a pair of meshed non-circular gears one of which is connected to each ofsaid rotors, said gears having a plurality of identical sectors, each ofsaid gear sectors having a pitch diameter in the form of a logarithmicspiral, said gears being meshed so that said gears contact at differentpoints along said respective pitch diameters of said sectors wherebysaid gears and said rotors rotate at varying speeds with respect to eachother to cause said pistons to alternately approach and recede.

9. Apparatus in accordance with claim 8 wherein each of said rotors hasfour pistons connected thereto so as to define eight of said chambers.

10. Apparatus in accordance with claim 8 including intermediate gearsfor interconnecting said non-circu lar gears with respective ones ofsaid rotors so that said rotors can rotate at speeds different from saidnon-circular gears.

11. Apparatus in accordance with claim 8 wherein each of said gears hasfour sectors forming four identical quadrants for said gears, said gearsbeing meshed so that said pistons approach and recede from each of theadjacent pistons one time thereby undergoing -a completecontraction-expansion cycle during each 180 degrees of rotation of saidgears. I

12. Apparatus in accordance with claim 8 wherein said logarithmic spiraldefining said pitch diameter of said sectors is given by the equation:

where: D is the radius from the center of said gear to any point alongsaid pitch diameter;

2 is the base of the natural logarithm;

a is the angle between the radius to the starting point of said pitchdiameter and said radius D; and

k is a constant determined by the ratio of the radii to the ending pointand the starting point of said pitch diameter.

13. Apparatus in accordance with claim 8 wherein said housing isconnected to one of said rotors and rotates therewith, said housingincluding a first intermediate gear on the exterior thereof;

a second intermediate gear identical to said first intermediate gear andmeshing therewith, said second intermediate gear being connected to oneof said meshed non-circular gears to rotate therewith so that saidrotors rotate in the same direction.

14. Apparatus in accordance with claim 13 including means for sealingsaid housing so that said chambers are isolated from each other, saidmachine including inlet and outlet ports therein for permitting movementof substance into and out of said chambers as said chambers expand andcontract.

15. Apparatus in accordance with claim 8 wherein said gear has foursectors comprising four identical gear quadrants, said quadrants beingsymmetrical with respect to two orthogonal axes through the center ofsaid gear, said gears being meshed such that they are rotated withrespect to each other when the mesh point thereof is on either of saidaxes so that said gears undergo one complete oscillating cycle withrespect to each other during each of rotation.

16. Apparatus in accordance with claim 15 wherein said logarithmicspiral is given by the equation:

where: D is the radius from the center of said gear to any point alongsaid pitch diameter of said quadrant;

e is the base of the natural logarithm;

a is the angle between the radius to the starting point of said pitchdiameter and said radius D; and

k is a constant determined by the ratio of the radii to the ending pointand said starting point of said pitch diameter, said ratio having avalue of four so that said chambers expand and contract approximately 30during said oscillating cycle.

1. A pair of meshed non-circular gears each of which has a plurality ofgear sectors, each of said sectors having a pitch diameter defined by alogarithmic spiral, said gears being meshed so that said gears contacteach other at different relative points along said respective pitchdiameters as said gears rotate whereby one of said gears alternatelyrotates at a faster and then a slower rate than the other of said gears.2. A pair of meshed non-circular gears for causing a first rotorconnected to one of said gears alternately to rotate at a faster andthen a slower rate than a second rotor connected to the other of saidgears thereby to produce an oscillating rotary motion between saidrotors, each of said gears having a plurality of identical sectors eachof which has a pitch diameter defined by a logarithmic spiral of theform:D eka, where: D is the radius from the center of said gear to anypoint along said pitch diameter; e is the base of the natural logarithm;a is the angle between the radius to the starting point of said pitchdiameter and said radius D; and k is a constant determined by the ratioof the radii to the ending point and said starting point of said pitchdiameter, said gears being rotated with respect to each other beforemeshing so that said gears contact each other at different relativepoints along said pitch diameters as said gears rotate, whereby saidoscillating rotary motion is obtained.
 3. Apparatus in accordance withclaim 2 wherein said ratio of said radii is four whereby saidoscillating motion of said rotors with respect to each other isapproximately 30*.
 4. Apparatus in accordance with claim 2 wherein thenumber of teeth and tooth spaces in each of said sectors is the same andeach is an integer plus one half so that adjacent ones of said sectorscan be joined to form a continuous set of teeth and spaces.
 5. Apparatusin accordance with claim 2 including intermediate gears interconnectingsaid noN-circular gears with respective ones of said rotors so that saidrotors can rotate at speeds different than said non-circular gears. 6.Apparatus in accordance with claim 2 wherein said gear has four sectorsforming four gear quadrants, said quadrants being symmetrical about eachof two orthogonal axes through the center of said gear, said gears beingmeshed so that one complete oscillating cycle of said rotors is obtainedas said gears rotate through two of said quadrants corresponding to 180*of rotation.
 7. Apparatus in accordance with claim 6 wherein said gearsare rotated with respect to each other so that said starting point ofone of said gears meshes with said ending point of the other of saidgears.
 8. In a rotary piston machine including a housing, a pair ofcoaxial rotors within said housing, and an equal number of pistonsconnected to each of said rotors for movement therewith within saidhousing, said pistons on said rotors alternating about the circumferenceof said housing and defining a plurality of chambers therein; means forinterconnecting said rotors and adapted to cause adjacent ones of saidpistons to alternately approach and recede from each other as saidrotors rotate within said housing thereby to cause said chambers toalternately contract and expand, including: a pair of meshednon-circular gears one of which is connected to each of said rotors,said gears having a plurality of identical sectors, each of said gearsectors having a pitch diameter in the form of a logarithmic spiral,said gears being meshed so that said gears contact at different pointsalong said respective pitch diameters of said sectors whereby said gearsand said rotors rotate at varying speeds with respect to each other tocause said pistons to alternately approach and recede.
 9. Apparatus inaccordance with claim 8 wherein each of said rotors has four pistonsconnected thereto so as to define eight of said chambers.
 10. Apparatusin accordance with claim 8 including intermediate gears forinterconnecting said non-circular gears with respective ones of saidrotors so that said rotors can rotate at speeds different from saidnon-circular gears.
 11. Apparatus in accordance with claim 8 whereineach of said gears has four sectors forming four identical quadrants forsaid gears, said gears being meshed so that said pistons approach andrecede from each of the adjacent pistons one time thereby undergoing acomplete contraction-expansion cycle during each 180 degrees of rotationof said gears.
 12. Apparatus in accordance with claim 8 wherein saidlogarithmic spiral defining said pitch diameter of said sectors is givenby the equation: D eka, where: D is the radius from the center of saidgear to any point along said pitch diameter; e is the base of thenatural logarithm; a is the angle between the radius to the startingpoint of said pitch diameter and said radius D; and k is a constantdetermined by the ratio of the radii to the ending point and thestarting point of said pitch diameter.
 13. Apparatus in accordance withclaim 8 wherein said housing is connected to one of said rotors androtates therewith, said housing including a first intermediate gear onthe exterior thereof; a second intermediate gear identical to said firstintermediate gear and meshing therewith, said second intermediate gearbeing connected to one of said meshed non-circular gears to rotatetherewith so that said rotors rotate in the same direction. 14.Apparatus in accordance with claim 13 including means for sealing saidhousing so that said chambers are isolated from each other, said machineincluding inlet and outlet ports therein for permitting movement ofsubstance into and out of said chambers as said chambers expand andcontract.
 15. Apparatus in accordance with claim 8 wherein said gear hasfour sectors comprising four identical gear quadrants, said quadrantsbeing symmetrical wiTh respect to two orthogonal axes through the centerof said gear, said gears being meshed such that they are rotated 90*with respect to each other when the mesh point thereof is on either ofsaid axes so that said gears undergo one complete oscillating cycle withrespect to each other during each 180* of rotation.
 16. Apparatus inaccordance with claim 15 wherein said logarithmic spiral is given by theequation: D eka, where: D is the radius from the center of said gear toany point along said pitch diameter of said quadrant; e is the base ofthe natural logarithm; a is the angle between the radius to the startingpoint of said pitch diameter and said radius D; and k is a constantdetermined by the ratio of the radii to the ending point and saidstarting point of said pitch diameter, said ratio having a value of fourso that said chambers expand and contract approximately 30* during saidoscillating cycle.